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Commutators of Bilinear θ-Type Calderón–Zygmund Operators on Morrey Spaces Over Non-Homogeneous Spaces

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Abstract

The aim of this paper is to establish the boundedness of the commutator [b1, b2, Tθ], which generated by the bilinear θ-type Calderón–Zygmund operators Tθ and the functions \(b_1, b_2 \in \widetilde {RBMO}(\mu)\), on non-homogeneous metric measure space satisfying the so-called geometrically doubling and the upper doubling conditions. Under the assumption that the dominating function λ satisfies the ε-weak reverse doubling conditions, the author proves that the commutator [b1, b2, Tθ] is bounded from the Lebesgue space Lp(μ) into the product of Lebesgue space \({L^{{p_1}}}(\mu ) \times {L^{{p_2}}}(\mu )\) with \(\frac{1}{p} = \frac{1}{{{p_1}}} + \frac{1}{{{p_2}}}(1 < p,{p_1},{p_2} < \infty )\). Furthermore, the boundedness of the commutator [b1, b2, Tθ] on Morrey space \(M_p^q(\mu)\) is also obtained, where 1 < q ≤ p < ∞.

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References

  1. T. A. Bui and X. T. Duong, Hardy spaces, regularized BMO spaces and the boundedness of Calderón–Zygmund operators on non-homogeneous spaces, J. Geom. Anal., 23 (2013), 895–932.

    Article  MathSciNet  Google Scholar 

  2. J. Chen and H. Lin, Maximal multilinear commutators on Non-homogeneous metric measure spaces, Taiwanese J. Math., 21 (2017), 1133–1160.

    Article  MathSciNet  Google Scholar 

  3. W. Chen, Y. Meng and D. Yang, Calderón–Zygmund operator on Hardy spaces without the doubling condition, Proc. Amer. Math. Soc., 133 (2005), 2671–2680.

    Article  MathSciNet  Google Scholar 

  4. W. Chen and E. Sawyer, A note on commutators of fractional integrals with RBMO(μ) functions, Illinois J. Math., 46 (2002), 1287–1298.

    Article  MathSciNet  Google Scholar 

  5. Y. Cao and J. Zhou, Morrey spaces for nonhomogeneous metric measure spaces, Abstr. Appl. Anal. (2013), Article ID 196459, 8 pp.

    Google Scholar 

  6. R. R. Coifman and G. Weiss, Analyse Harmonique Non-Commutative sur Certains Espaces Homogènes, Lecture Notes in Mathematics, vol. 242, Springer-Verlag (Berlin–New York, 1971).

    Book  Google Scholar 

  7. R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83 (1977), 569–645.

    Article  MathSciNet  Google Scholar 

  8. X. Fu, Da. Yang and Do. Yang, The molecular characterization of the Hardy space H1 on non-homogeneous metric measure spaces and its application, J. Math. Anal. Appl., 410 (2014), 1028–1042.

    Article  MathSciNet  Google Scholar 

  9. X. Fu, D. Yang and W. Yuan, Boundedness of multilinear commutators of Calderón–Zygmund operators on Orlicz spaces over non-homogeneous spaces, Taiwanese J. Math., 16 (2012), 2203–2238.

    Article  MathSciNet  Google Scholar 

  10. X. Fu, D. Yang and W. Yuan, Generalized fractional integral and their commutators over non-homogeneous metric measure spaces, Taiwanese J. Math., 18 (2014), 509–557.

    Article  MathSciNet  Google Scholar 

  11. T. Hyt¨onen, A framework for non-homogeneous analysis on metric spaces, and the RBMO(μ) space of Tolsa, Publ. Math., 54 (2010), 485–504.

    Article  MathSciNet  Google Scholar 

  12. G. Hu, Y. Meng and D. Yang, New atomic characterization of H1 space with nondoubling measures and its applications, Math. Proc. Cambridge Philos. Soc., 138 (2005), 151–171.

    Article  MathSciNet  Google Scholar 

  13. T. Hyt¨onen, Da. Yang and Do. Yang, The Hardy space H1 on non-homogeneous metric spaces, Math. Proc. Cambridge Philos. Soc., 153 (2012), 9–31.

    Article  MathSciNet  Google Scholar 

  14. H. Lin, S. Wu and D. Yang, Boundedness of certain commutators over nonhomogeneous metric measure spaces, Anal. Math. Phys., 7 (2017), 1–32.

    Article  MathSciNet  Google Scholar 

  15. G. Lu and S. Tao, Bilinear θ-type Calderón–Zygmund operators non-homogeneous metric measure spaces, J. Comp. Anal. Appl., 26 (2019), 650–670.

    Google Scholar 

  16. G. Lu and S. Tao, Generalized Morrey spaces over non-homogeneous metric measure spaces, J. Aust. Math. Soc., 103 (2017), 268–278.

    Article  MathSciNet  Google Scholar 

  17. G. Lu and S. Tao, Commutators of Littlewood–Paley g*κ-functions on nonhomogeneous metric measure spaces, Open Math., 15 (2017), 1283–1299.

    MathSciNet  MATH  Google Scholar 

  18. G. Lu and J. Zhou, Estimates for fractional type Marcinkiewicz integrals with nondoubling measures, J. Inequal. Appl., 2014, 2014:285, 14 pp.

    Article  Google Scholar 

  19. F. Nazarov, S. Treil and A. Volberg, Weak type estimates and Cotlar inequalities for Calderón–Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices 1998, 463–487.

    Google Scholar 

  20. C. Ri and Z. Zhang, Boundedness of θ-type Calderón–Zygmund operators on Hardy spaces with non-doubling measures, J. Inequal. Appl., 2015, 2015:323, 10 pp.

    Article  Google Scholar 

  21. C. Ri and Z. Zhang, Boundedness of θ-type Calderón–Zygmund operators on nonhomogeneous metric measure spaces, Front. Math. China, 11 (2016), 141–153.

    Article  MathSciNet  Google Scholar 

  22. Y. Sawano and H. Tanaka, Morrey spaces for non-doubling measures, Acta Math. Sin. (Engl. Ser.), 21 (2005), 1535–1544.

    Article  MathSciNet  Google Scholar 

  23. Y. Sawano and H. Tanaka, Sharp maximal inequalities and commutators on Morrey spaces with non-doubling measures, Taiwanese J. Math., 11 (2007), 1091–1112.

    Article  MathSciNet  Google Scholar 

  24. X. Tolsa, BMO, H1 and Calderón-Zygmund operators for non-doubling measure, Math. Ann., 319 (2001), 142–158.

    Article  Google Scholar 

  25. X. Tolsa, Littlewood-Paley theory and T(1) theorem with non-doubling measures, Adv. Math., 164 (2001), 57–116.

    Article  MathSciNet  Google Scholar 

  26. X. Tolsa, The space H1 for non-doubling measure spaces in terms of a grand maximal operator, Trans. Amer. Math. Soc., 355 (2003), 315–348.

    Article  MathSciNet  Google Scholar 

  27. R. Xie and L. Shu, Θ-type Calderón-Zygmund operators with non-doubling measures, Acta Math. Appl. Sin. (Engl. Ser.), 29 (2013), 263–280.

    Article  MathSciNet  Google Scholar 

  28. R. Xie, L. Shu and A. Sun, Boundedness for commutators of bilinear θ-type Calderón-Zygmund operators on non-homogheneous metric measure spaces, J. Func. Spaces, 2017, Art. ID 3690452, 10 pp.

  29. K. Yabuta, Generalization of Calderón-Zygmund operators, Studia Math., 82 (1985), 17–31.

    Article  MathSciNet  Google Scholar 

  30. T. Zheng, X. Tao and X. Wu, Bilinear Calderón-Zygmund operators of type ω(t) on non-homogeneous space, Ann. Funct. Anal., 6 (2015), 134–154.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu, 730070, P. R. China

    G.-H. Lu

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  1. G.-H. Lu
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Correspondence to G.-H. Lu.

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This research was supported by the Doctoral Scientific Research Foundation of Northwest Normal University (No. 0002020203).

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Lu, GH. Commutators of Bilinear θ-Type Calderón–Zygmund Operators on Morrey Spaces Over Non-Homogeneous Spaces. Anal Math 46, 97–118 (2020). https://doi.org/10.1007/s10476-020-0020-3

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  • DOI: https://doi.org/10.1007/s10476-020-0020-3

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